Guided Proof Let v1,v2,....Vn be a basis for a vector space V. Prove that if...

facas9

facas9

Answered

2021-01-17

Guided Proof Let v1,v2,....Vn be a basis for a vector space V.
Prove that if a linear transformation T:VV satisfies
T(vi)=0 for i=1,2,...,n, then T is the zero transformation.
To prove that T is the zero transformation, you need to show that T(v)=0 for every vector v in V.
(i) Let v be the arbitrary vector in V such that v=c1v1+c2v2++cnVn
(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of T(vj) .
(iii) Use the fact that T(vj)=0
to conclude that T(v)=0, making T the zero transformation.

Answer & Explanation

sweererlirumeX

sweererlirumeX

Expert

2021-01-18Added 91 answers

a)Given:
The linear transformation T:VV
represented as T(vi)=0 for i=1,2,...,n.
Approach:
Consider an arbitrary v=v1,v2,...,vn is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
T(vi)=0,i=1,2,,n...(1)
Calculation:
As the vector set v is the subspace of V, the vector v can be written linear combination.
Write the subspace vas linear combination.
v=c1v1+c2v2++cnvn,(2)
Here, c1,c2,cn are arbitrary scalars.
Conclusion:
Hence, it is proved above that the set (v1,v2,vn) is represented as
v=c1n1+c2v2,++cnvn.
b)Given:
The linear transformation T:VV
represented as T(vi)=0 for i=1,2,,n.
Approach:
Consider an arbitrary v=v1,v2,,vn is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
T(vi)=0,i=1,2,,n(1)
The vector additinon is given by,
T(u+v)=T(u)+T(v)
The scalar multiplication is given by,
T(cu)=cT(u)
Calculation:
As the vector set v is the subspace of V, the vector v can be written linear combination.
Write the subspace vas linear combination.
T(v)=(c1v1+c2v2++cnvn)
=T(c1v1)+T(c2v2)++T(cnvn)
=c1T(v1)+c2T(v2)++cnT(vn)....(3)
Conclusion:
The transformation form of linear combination v=c1v1+c2v2++cnvn is
T(v)=c1T(v1)+c2T(v2)++cnT(vn)
c) Given:
The linear transformation T:VV
represented as T(vi)=0 for i=1,2,,n.
Approach:
Consider an arbitrary v=v1,v2,,vn is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
T(vi)=0,i=1,2,,n(1)
The vector additinon is given by,
T(u+v)=T(u)+T(v)
The scalar multiplication is given by,
T(cu)=cT(u)
Calculation:
Solve formula (3) with use of formula(1)
T(v)=c1T(v1)+c2T(v2)++cnT(vn)
=c1(0)+c2(0)++cn(0)
= 0
From above calculation is is clear linear transformation T:VV
satisfies

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